Paths with many shortcuts in tournaments

TL;DR

This paper investigates paths with many shortcuts in tournaments, proving the existence of Hamiltonian paths with numerous hops and hop complete paths, and analyzes the maximum number of shortcuts in spanning shortcut trees.

Contribution

It establishes lower bounds on the number of hops and shortcuts in paths and trees within tournaments, advancing understanding of their structural properties.

Findings

Every tournament has a Hamiltonian path with at least (4n-10)/7 hops.

Existence of a hop complete path of order at least n^{0.295}.

Asymptotic growth of maximum shortcuts in spanning shortcut trees is Θ(n log^2 n).

Abstract

A shortcut of a directed path $v_1 v_2 \cdots v_n$ is an edge $v_iv_j$ with $j > i+1$. If $j = i+2$ the shortcut is called a hop. If all hops are present, the path is called hop complete, so the path and its hops form a square of a path. We prove that every tournament with $n \ge 4$ vertices has a Hamiltonian path with at least $(4n-10)/7$ hops, and has a hop complete path of order at least $n^{0.295}$. A spanning binary tree of a tournament is a spanning shortcut tree if for every vertex of the tree, all its left descendants are in-neighbors and all its right descendants are out-neighbors. It is well-known that every tournament contains a spanning shortcut tree. The number of shortcuts of a shortcut tree is the number of shortcuts of its unique induced Hamiltonian path. Let $t(n)$ denote the largest integer such that every tournament with $n$ vertices has a spanning shortcut tree with at least $t(n)$ shortcuts. We almost determine the asymptotic growth of $t(n)$ as it is proved that $\Theta(n\log^2n) \ge t(n)-\frac{1}{2}\binom{n}{2} \ge \Theta(n \log n)$.